find the zeros of the function:
f(x)=x^3+2x^2-109x-110
I think I need to factor but everytime I do, it doesn't look right.
To find the zeros of the given function, we can use a few different methods, such as factoring, synthetic division, or using the rational root theorem. Let's try using the rational root theorem to narrow down the possible zeros first.
The rational root theorem states that if a polynomial equation has any rational roots, they must be in the form of p/q, where p is a factor of the constant term (in this case, -110) and q is a factor of the leading coefficient (in this case, 1).
The factors of the constant term (-110) are: ±1, ±2, ±5, ±10, ±11, ±22, ±55, ±110.
The factors of the leading coefficient (1) are: ±1.
Now, we can systematically test each possible combination of p and q to check if any of these values yield a zero for the function f(x)=x^3+2x^2-109x-110.
Let's start with p/q = 1:
If we substitute x = 1 into the function f(x), we get:
f(1) = (1)^3 + 2(1)^2 - 109(1) - 110 = 1 + 2 - 109 - 110 = -216.
Since f(1) is not zero, x = 1 is not a zero of the function.
Let's continue testing the remaining values from the rational root theorem until we find a zero.
After systematically testing all the possible rational roots using the rational root theorem, we find that x = -10 is a zero of the given function f(x)=x^3+2x^2-109x-110.
Now, we can use synthetic division or long division to divide the polynomial by (x+10) to find the quadratic factor and further factor the equation.
Performing synthetic division with x = -10 gives the following result:
-10 | 1 2 -109 -110
|
|_________-10_____80____0__
1 -8 -29 -110
The quadratic factor is x^2 - 8x - 29.
Factoring the quadratic factor (x^2 - 8x - 29) can be done using the quadratic formula or by finding two numbers that multiply to -29 and add up to -8. These numbers are -2 and -14.
Therefore, the zeros of the given function f(x)=x^3+2x^2-109x-110 are:
x = -10 (found using the rational root theorem)
x = -2
x = -14
you need to put this in synthetic division
take the coef from power of greatest to least
1 2 -109 -110 and use factors of -110 to try
-1 works so (X+1) is a factor of equation
rewrite: (x+1)(X^2+X-110)
find root of (X^2+X-110) COME TO be -11,10
so factor form is (X+11)(X-10)(X+1)
n find the roots